API usage and design overview

In ths document we give an overview on the structure
of the interfaces, classes and packages of JAS.
In the first section we show how to compute Legendre polynomials
with the JAS API.
In the next three sections we focus on the structure
of the required types and the creation of the corresponding objects.
In the following three sections we focus on the functional aspects of
the types, i.e. their constructors and methods.
For a discussion of other design alternatives see
the problems document.
Further programming issues and bugs are listed in the
Findbugs report.

1. Getting started

You can use whatever environment for programming in Java you like. The
next picture shows the usage of Eclipse, but any text editor and a
Java compiler will do as well.

JAS program to compute Legendre polynomials in the Eclipse IDE

1.1. Computation of the Legendre polynomials

At first we present an example for the usage of the JAS API
with the computation of the Legendre polynomials.
The Legendre polynomials can be defined by the following recursion

The polynomials have been computed with the following Java program.
First we need a polynomial ring ring over the rational numbers,
in one variable "x" and a list P to store the
computed polynomials.
The polynomial factory object itself needs at least a factory for
the creation of coefficients and the number of variables.
Additionally the term order and names for the variables can be
specified. With this information the polynomial ring factory
can be created by
new GenPolynomialRing <BigRational>(fac,1,var),
where fac is the coefficient factory,
1 is the number of variables, and
var is an String array of names.

The polynomials for the recursion base are one and x.
Both are generated from the polynomial ring factory
with method ring.getONE() and ring.univariate(0),
respectively.
The polynomial (2n-1)x is produced in the for-loop by
n21 = new BigRational( 2*i-1 ); and
xc = x.multiply( n21 );.
The polynomial (n-1) P[n-2] is computed by
nn = new BigRational( i-1 ); and
xc = P.get(i-2).multiply( nn ).
Finally we have to multiply the difference of the
intermediate polynomials by 1/i as
nn = new BigRational( 1, i ); and
t = t.multiply( nn ).
Then, in the for-loop, the polynomials P[i] are computed
using the definition, and stored in the list P for further use.
In the last for-loop, the polynomials are printed, producing the
output shown above.
The string representation of the polynomial object can be created,
as expected, by toString(),
or by using names for the variables with toString(var).
The imports required are

To use other coefficient rings, one simply changes the
generic type parameter, say, from BigRational to
BigComplex and adjusts the coefficient factory.
The factory would then be created as
c = new BigComplex(), followed by
new GenPolynomialRing<BigComplex> (c,1,var).
This small example shows that this library can easily be used,
just as any other Java package or library.

In the following sections we describe the
central classes and interfaces for the polynomial API.

1.2. Algebraic structures overview

To get an idea of the scope of JAS we summarize
the implemented algebraic structures and
of the implemented algebraic algorithms.

class

factory

structure

methods

BigInteger

self

ring of arbitrary precision integers,
a facade for java.math.BigInteger

ring of recursive, solvable polynomials in r variables
over a solvable polynomial coefficient ring over any implemented coefficient ring
with respect to any implemented term ordering
(compatible with the multiplication).
The other variants have coefficient rings SolvableQuotient,
SolvableResidue and SolvableLocal which are all based
on solvable polynomials.
The QLRSolvablePolynomial is a unification of the others.

1.3. Algebraic algorithms overview

Iterated subtraction of polynomials to eliminate terms from a given polynomial,
i.e. reduction of polynomial(s) wrt. a set of polynomials.
Coefficients of polynomials must be from a field and for the
Pseudo* version from a ring with gcd.
For *Par the list of polynomials can concurrently be modified.

normalform,
S-polynomial,
criterions,
extended normalform

DReduction,
EReduction,
DReductionSeq,
EReductionSeq

Reduction of polynomial(s) wrt. a set of polynomials.
Coefficients of polynomials must be from a
principial ideal domain (PID) or from an Euclidean domain.

Iterated subtraction of polynomials to eliminate terms from a given polynomial,
i.e. reduction of polynomial(s) wrt. a set of polynomials.
Coefficients of polynomials must be from a regular ring and for the
Pseudo* version from a regular ring with gcd.
Boolean closure and boolean remainder of polynomials.

normalform,
S-polynomial,
boolean closure

CReductionSeq,
Condition,
ColorPolynomial

Iterated subtraction of polynomials to eliminate terms from a given polynomial,
i.e. reduction of polynomial(s) wrt. a set of polynomials.
Coefficients of polynomials must be from a polynomial ring.
Case distinction and determination of polynomaials with respect to conditions
leading to colored polynomials.

Buchberger algorithm to compute Groebner bases
of sets of polynomials.
Coefficients of polynomials must be from a field.
*Parallel is a multi-threaded and
*Distributed is a message passing implementation.
The *Pseudo versions are for coefficient domains with gcd.

GB, isGB, extended GB, minimal GB

DGroebnerBaseSeq,
EGroebnerBaseSeq

Algorithm to compute D- and E- Groebner bases
of sets of polynomials.
Coefficients of polynomials must be from a
principial ideal domain (PID) or from an Euclidean domain.

Algorithm to compute left, right and two-sided Groebner bases
of sets of solvable polynomials.
Coefficients of polynomials must be from a field.
Parallel is a multi-threaded implementation.
The *Pseudo versions are for coefficient domains with gcd.

Algorithm to compute two-sided Groebner bases
of sets of free non-commutative polynomials.
Coefficients of polynomials must be from a field.
The *Pseudo versions are for coefficient domains with gcd.

two-sided versions of GB, isGB, minimal GB

RGroebnerBaseSeq,
RGroebnerBasePseudoSeq

Algorithm to compute Groebner bases in polynomial rings over regular rings.
Coefficients of polynomials must be from a
product of fields or Euclidean domains.

GB, isGB, minimal GB

ComprehensiveGroebnerBaseSeq,
GroebnerSystem,
ColoredSystem

Algorithm to compute comprehensive Groebner bases in polynomial rings over
parameter rings.
Coefficients of polynomials must be from a polynomial ring.
Computation is done via Groebner systems (lists of colored systems).

GBsys, isGBsys, GB, isGB, minimalGB

Syzygy,
SyzygyAbstract,
ModGroebnerBase,
ModGroebnerBaseAbstract

Algorithm to compute syzygies of lists of polynomials
or Groebner Bases, free resolutions.
Groebner Bases for modules over polynomial rings.
Coefficients of polynomials must be from a field.

Algorithm to compute left and right syzygies of lists
of solvable polynomials or Groebner Bases, free left resolutions.
Left, right and two-sided Groebner Bases for modules over solvable polynomial rings.
Coefficients of polynomials must be from a field.

Mora's tangent cone reduction algorithm and computation of standard bases
of sets of multivariate power series.
Coefficients of polynomials must be from a field.

STD, isSTD, minimalSTD, normalForm, SPolynomial

Ideal

Algorithms to compute sums, products, intersections, containment
and (infinite) quotients of polynomial ideals.
Coefficients of polynomials must be from a field.
Prime, primary, irreducible and radical decomposition of zero dimensional ideals.
Prime, primary, irreducible and radical decomposition of non-zero dimensional ideals.
Univariate polynomials of minimal degree in ideal as well as
elimination, extension and contraction ideals.

Algorithms to compute greatest common divisors of polynomials
via different polynomial remainder sequences (PRS) and modular methods.
Coefficients of polynomials must be from a unique factorization domain (UFD).
GCDFactory helps with the optimal selection of an algorithm
and GCDProxy uses multi-threading to compute with
several implementations in parallel.

Algorithms to compute squarefree decomposition of polynomials
over fields of characteristic zero, finite and infinite fields of characteristic p
and other coefficients from unique factorization domains (UFD).
SquarefreeFactory helps with the optimal selection of an algorithm.

Algorithms to compute factorizations of polynomials as products of irreducible polynomials
over different ground rings.
FactorFactory helps with the correct selection of an algorithm.
Reduction of the multivariate factorization to an univariate factorization is done
with Kronecker's algorithm in the general case and with Wang's algorithm over the integers.

Algorithms to compute isolating intervals for real roots
and for refinement of isolating intervals to any prescribed precision.
Algorithms to compute the sign of a real algebraic numer and
the magnitude of a real algebraic number to a given precision.
Coefficients of polynomials must be from a real field,
for example from BigRational or RealAlgebricNumber.

realRoots, refineInterval, algebraicSign, algebraicMagnitude

ComplexRoots,
ComplexRootsAbstract,
ComplexRootsSturm

Algorithms to compute isolating rectangles for complex roots
and for refinement of isolating rectangles to any prescribed precision.
Coefficients of polynomials must be of type Complex field.

1.4. Packages and components overview

2. Recursive ring element design

The next figure gives an overview of the central interfaces and classes.
The interface RingElem defines a recursive type
which defines the functionality (see next section) of the
polynomial coefficients and is also implemented by the
polynomials itself. So polynomials can be taken as coefficients
for other polynomials, thus defining a recursive polynomial ring structure.

Since the construction of constant ring elements has been difficult
in previuos designs, we separated the creational aspects of ring elements
into ring factories with sufficient context information.
The minimal factory functionality is defined by the interface
RingFactory.
Constructors for polynomial rings will then require factories for
the coefficients so that the construction of polynomials over these
coefficient rings poses no problem.
The ring factories are additionaly required because of the
Java generic type design. I.e. if C is a generic type name
it is not possible to construct an new object with new C().
Even if this would be possible, one can not specify constructor signatures
in Java interfaces, e.g. to construct a one or zero constant ring element.
Recursion is again achieved by using polynomial factories as
coefficient factories in recursive polynomial rings.
Constructors for polynomials will always require
a polynomial factory parameter which knows
all details about the polynomial ring under consideration.

UML diagram of JAS types

3. Coefficients and polynomials

We continue the discussion of the next layer of classes in the
the above figure.

Elementary coefficient classes, such as
BigRational or BigInteger, implement both
the RingElem and RingFactory interfaces.
This is convenient, since these factories do not need further context
information.
In the implementation of the interfaces the type parameter
C extends RingElem<C> is simultaneously bound
to the respective class, e.g. BigRational.
Coefficient objects can in most cases created directly
via the respective class constructors, but also via the factory methods.
E.g. the object representing the rational number 2 can be created by
new BigRational(2) or by
fac = new BigRational(), fac.fromInteger(2) and
the object representing the rational number 1/2 can be created by
new BigRational(1,2) or by
fac.parse("1/2").

Generic polynomials are implemented in the
GenPolynomial class, which has a type parameter
C extends RingElem<C>
for the coefficient type. So all operations on coefficients required
in polynomial arithmetic and manipulation are guaranteed to exist by the
RingElem interface. The constructors of the polynomials
always require a matching polynomial factory.
The generic polynomial factory is implemented in the class
GenPolynomialRing, again with type parameter
C extends RingElem<C> (not RingFactory).
The polynomial factory however implements the interface
RingFactory<C extends RingElem<C>> so that
it can also be used recursively.
The constructors for GenPolynomialRing require at least
parameters for a coefficient factory and the number of variables
of the polynomial ring.

Having generic polynomial and elementary coefficient implementations
one can attempt to construct polynomial objects.
The type is first created by binding the type parameter
C extends RingElem<C> to the desired coefficient type,
e.g. BigRational. So we arrive at the type
GenPolynomial<BigRational>.
Polynomial objects are then created via the respective polynomial factory
of type GenPolynomialRing<BigRational>,
which is created by binding the generic coefficient type of the
generic polynomial factory to the desired coefficient type,
e.g. BigRational.
A polynomial factory object is created from a
coefficient factory object and the number of variables in the
polynomial ring as usual with the new operator via one
of its constructors.
Given an object coFac of type BigRational,
e.g. created with new BigRational(), a polynomial factory
object pfac of the above described type could be created by
new GenPolynomialRing<BigRational>(coFac,5).
I.e. we specified a polynomial ring with 5 variables over
the rational numbers.
A polynomial object p of the above described type can then
be created by any method defined in RingFactory,
e.g. by pfac.getONE(),
pfac.fromInteger(1), pfac.random(3)
or pfac.parse("(1)").

Since GenPolynomial itself implements the
RingElem interface, they can also be used recursively
as coefficients.
We continue the polynomial example and are going to use polynomials over
the rational numbers as coefficients of a new polynomial.
The type is then
GenPolynomial<GenPolynomial<BigRational>>
and the polynomial factory has type
GenPolynomialRing<GenPolynomial<BigRational>>.
Using the polynomial coefficient factory pfac from above
a recursive polynomial factory rfac could be created by
new GenPolynomialRing<GenPolynomial<BigRational>>(pfac,3).
The creation of a recursive polynomial object r of the
above described type is then as a easy as before
e.g. by rfac.getONE(),
rfac.fromInteger(1) or rfac.random(3).

4. Solvable polynomials

We turn now to the last layer of classes in the the above figure.

The generic polynomials are intended as super class for further
types of polynomial rings. As one example we take so called
solvable polynomials, which are like normal polynomials
but are equipped with a new non-commutative multiplication.
They are implemented in the class GenSolvablePolynomial
which extends GenPolynomial and inherits all methods
except clone() and multiply().
The class also has a type parameter C extends RingElem<C>
for the coefficient type.
Note, that the inherited methods are in fact creating solvable polynomials
since they employ the solvable polynomial factory for the creation of any
new polynomial internally. Only the formal method return type is that
of GenPolynomial, the run-time type is
GenSolvablePolynomial to which they can be casted at any time.
The factory for solvable polynomials is implemented
by the class GenSolvablePolynomialRing which also
extends the generic polynomial factory. So this factory can also be used
in the constructors of GenPolynomial via super()
to produce in fact solvable polynomials internally. The data structure
is enhanced by a table of non-commutative relations defining the
new multiplication. The constructors delegate most things to the
corresponding super class constructors and additionally have a
parameter for the RelationTable to be used.
Also the methods delegate the work to the respective super class methods
where possible and then handle the non-commutative multiplication relations
separately.

The construction of solvable polynomial objects
follows directly that of polynomial objects.
The type is created by binding the type parameter
C extends RingElem<C> to the desired coefficient type,
e.g. BigRational. So we have the type
GenSolvablePolynomial<BigRational>.
Solvable polynomial objects are then created via the respective
solvable polynomial factory of type
GenSolvablePolynomialRing<BigRational>,
which is created by binding the generic coefficient type of the
generic polynomial factory to the desired coefficient type,
e.g. BigRational.
A solvable polynomial factory object is created from a
coefficient factory object, the number of variables in the
polynomial ring and a table containing the defining non-commutative relations
as usual with the new operator via one of its constructors.
Given an object coFac of type BigRational
as before, a polynomial factory object spfac
of the above described type could be created by
new GenSolvablePolynomialRing<BigRational>(coFac,5).
I.e. we specified a polynomial ring with 5 variables over
the rational numbers with no commutator relations.
A solvable polynomial object p of the above described type
can then be created by any method defined in RingFactory,
e.g. by spfac.getONE(),
spfac.fromInteger(1), spfac.random(3)
or spfac.parse("(1)").
Some care is needed to create RelationTable objects
since its constructor requires the solvable polynomial ring which
is under construction as parameter. It is most convenient to first
create a GenSolvablePolynomialRing with an
empty relation table and then to add the defining relations.

5. Ring element and factory functionality

The following sections and the next figure gives an overview
of the functionality of the main interfaces and polynomial classes.

The RingElem interface has a generic type parameter
C which is constrained to a type with the same functionality
C extends RingElem<C>.
It defines the usual methods required for ring arithmetic such as
C sum(C S); C subtract(C S); C negate(); C abs();
C multiply(C S); C divide(C S); C remainder(C S); C inverse();
Although the actual ring may not have inverses for every element
or some division algorithm we have included these methods in the definition.
In a case where there is no such function, the implementation may
deliberately throw a RuntimeException or choose some
other meaningful element to return.
The method isUnit() can be used to check if an element
is invertible.

Besides the arithmetic method there are following testing methods
boolean isZERO(); boolean isONE(); boolean isUnit();
int signum();
boolean equals(Object b); int hashCode(); int compareTo(C b);
The first three test if the element is 0, 1 or a unit in the respective ring.
The signum() method defines the sign of the element (in case
of an ordered ring).
equals(), hashCode() and compareTo()
are required to keep Javas object machinery working in our sense.
They are used when an element is put into a Java collection class,
e.g. Set, Map or SortedMap.
The last method C clone() can be used to obtain a copy of the
actual element. As creational method one should better use the
method C copy(C a) from the ring factory, but in Java
it is more convenient to use the clone() method.

As mentioned before, the creational aspects of rings are separated
into a ring factory. A ring factory is intended to store all context
information known or required for a specific ring.
Every ring element should also know its ring factory, so all
constructors of ring element implementations require a parameter
for the corresponding ring factory. Unfortunately constructors
and their signature can not be specified in a Java interface.
The RingFactory interface also has a generic type parameter
C which is constrained to a type with the ring element
functionality C extends RingElem<C>.
The defined methods are
C getZERO(); C getONE();
C fromInteger(long a); C fromInteger(java.math.BigInteger a);
C random(int n); C copy(C c);
C parse(String s); C parse(Reader r);
The first two create 0 and 1 of the ring.
The second two are used to embed a natural number into the ring
and create the corresponding ring element.
The copy() method was intended as the main means to
obtain a copy of a ring element, but it is now no more used in our
implmentation. Instead the clone() method is used from
the ring element interface.
The random(int n) method creates a random element of the
respective ring. The parameter n specifies an appropriate
maximal size for the created element. In case of coefficients it
usually means the maximal bit-length of the element, in case of
polynomials it influences the coefficient size and the degrees.
For polynomials there are random() methods with more
parameters.
The two methods
C parse(String s) and C parse(Reader r)
create a ring element from some external string representation.
For coefficients this is mostly implemented directly and for
polynomials the class GenPolynomialTokenizer is
employed internally.
In the current implementation the external representation of
coefficients may never contain white space and must always start
with a digit.
In the future the ring factory will be enhanced by methods that
test if the ring is commutative, associative or has some
other important property or the value of a property,
e.g. is an euclidean ring, is a field, an integal domain,
a uniqe factorization domain, its characteristic or if it is noetherian.

UML diagram of JAS type functionality

6. Polynomial and polynomial factory functionality

We continue the discussion of the above figure
with the generic polynomial and factory classes.

The GenPolynomial class has a generic type parameter
C which is constrained to a type with the functionality
of ring elements C extends RingElem<C>.
Further the class implements a RingElem over itself
RingElem<GenPolynomial<C>>
so that it can be used for the coefficients of an other polynomial ring.
The functionality of the ring element methods has already been explained
in the previous section.
There are two public and one protected constructors, each requires
at least a ring factory parameter GenPolynomialRing<C> r.
The first creates a zero polynomial
GenPolynomial(. r),
the second creates a polynomial of one monomial with given coefficient
and exponent tuple
GenPolynomial(. r, C c, ExpVector e),
the third creates a polynomial from the internal sorted map of an
other polynomial
GenPolynomial(. r, SortedMap<ExpVector,C> v).
Further there are methods to access parts of the polynomial
like leading term, leading coefficient
(still called leading base coefficient from the Aldes/SAC-2 tradition)
and leading monomial.
The toString() method creates as usual a string representation
of the polynomials consisting of exponent tuples and coefficients.
One variant of it takes an array of variable names and creates a string
consisting of coefficients and products of powers of variables.
The method extend() is used to embed the polynomial into the
'bigger' polynomial ring specified in the first parameter.
The embeded polynomial can also be multiplied by a power of a variable.
The contract() method returns a map of exponents and
coefficients. The coefficients are polynomials belonging to the
'smaller' polynomial ring specified in the first parameter.
If the polynomial actually belongs to the smaller polynomial ring
the map will contain only one pair, mapping the zero exponent vector
to the polynomial with variables removed.
A last group of methods computes (extended) greatest common divisors.
They work correct for univariate polynomials over a field but not
for arbitrary multivatiate polynomials. These methods will be moved
to a new separate class in the future.

The GenPolynomialRing class has a generic type parameter
C which is constrained to a type with the functionality
of ring elements C extends RingElem<C>.
Further the class implements a RingFactory over
GenPolynomial<C> so that it can be used as
coefficient factory of a different polynomial ring.
The constructors require at least a factory for the coefficents as
first parameter of type RingFactory<C>
and the number of variables in the second parameter.
A third parameter can optionally specify a TermOrder
and a fourth parameter can specify the names for the variables
of the polynomial ring.
Besides the methods required by the RingFactory
interface there are additional random() methods
which provide more control over the creation of random polynomials.
They have the following parameters:
the bitsize of random coefficients to be used in the
random() method of the coefficient factory,
the number of terms (i.e. the length of the polynomial),
the maximal degree in each variable
and the density of nozero exponents, i.e. the ratio of nonzero to
zero exponents.
The toString() method creates a string representation
of the polynomial ring consisting of the coefficient factory string
representation, the tuple of variable names and the string representation
of the term order.
The extend() and contract() methods
create 'bigger' respectively 'smaller' polynomial rings.
Both methods take a parameter of how many variables are to be added
or removed form the actual polynomial ring.
extend() will setup an elimination term order consisting
of two times the actual term order when ever possible.

7. Solvable polynomial and solvable polynomial factory functionality

We continue the discussion of the above figure
with the generic solvable polynomial and factory classes.

The GenSolvablePolynomial class has a generic type parameter
C which is constrained to a type with the functionality
of ring elements C extends RingElem<C>.
The class extends the GenPolynomial class. It inherits
all additive functionality and overwrites the multiplicative
functionality with a new non-commutative multiplication method.
Unfortunately it cannot implement a RingElem
over itself RingElem<GenSolvablePolynomial<C>>
but can only inherit the implementation of
RingElem<GenPolynomial<C>> from its super class.
By this limitation a solvable polynomial can still be used as
coefficent in another polynomial, but only with the type of its super
class. The limitation comes form the erasure of template parameters in
RingElem<...> to RingElem for the
code generated. I.e. the generic interfaces become the same after
type erasure and it is not allowed to implement the same interface twice.
There are two public and one protected constructors as in the super class.
Each requires at least a ring factory parameter
GenSolvablePolynomialRing<C> r
which is stored in a variable of this type shadowing the variable
with the same name of the super factory type. The rest of the
initialization work is delegated to the super class constructor.

The GenSolvablePolynomialRing class has a generic type parameter
C which is constrained to a type with the functionality
of ring elements C extends RingElem<C>.
The class extends the GenPolynomialRing class.
It overwrites most methods to implement the new non-commutative
methods.
Also this class cannot implement a RingFactory over
GenSolvablePolynomial<C>. It only implements
RingFactory over GenPolynomial<C>
by inheritance by the same reason of type erasure as above.
But it can be used as coefficient factory with the type of its super class
for a different polynomial ring.
One part of the constructors just restate the super class constructors
with the actual solvable type. A solvable polynomial ring however
must know how to perform the non-commutative multiplication.
To this end a data structure with the respective commutator relations
is required. It is implemented in the RelationTable class.
The other part of the constructors additionaly takes a parameter of type
RelationTable to set the initial commutator relation table.
Some care is needed to create relation tables and solvable polynomial
factories since the relation table requires a solvable polynomial
factory as parameter in the constructor. So it is most advisable to
create a solvable polynomial factory object with empty relation table
and to fill it with commutator relations after the constructor is
completed but before the factory will be used.
There is also a new method isAssociative() which tries
to check if the commutator relations indeed define an associative
algebra. This method should be extracted to the RingFactory
interface together with a method isCommutative(),
since both are of general importance and not always fulfilled
in our rings. E.g. BigQuaternion is not commutative
and so is a polynomial ring over these coefficents is not commutative.
The same applies to associativity and the (not jet existing) class
BigOctonion.

This concludes the discussion of the main interfaces and classes
of the Java algebra system.